Cramer-type conjecture for values of binary quadratic forms representing sums of two squares

Let $Q(x,y):=Ax^2+2Bxy+Cy^2+Dx+Ey+F$ be a binary quadratic form with integral coefficients satisfying

$$\max(|A|,|B|,|C|,|E|,|F|)\leq M,$$

and with discriminant

$$\operatorname{Disc}(Q):=B^2-AC=q^2$$

for a prime $q=4k+1$. Suppose that $Q(x,y)$ is positive throughout the box $(x,y)\in[-M_1,M_1]\times[-M_2,M_2]$, where $M_1,M_2\geq1$. Cramer-type conjecture. There exists an exponent $\gamma\geq1/2$ and a constant $C_\gamma$, independent of $Q$, $q$, and the box, such that whenever

$$C_\gamma\log(M)^\gamma\leq M_1M_2,$$

$Q$ represents a sum of two squares at some point inside the box $[-M_1,M_1]\times[-M_2,M_2]$. This is proposed as an analogue of Cramer's conjecture for the distribution of numbers representable as sums of two squares; it is used to motivate a polylogarithmic algorithm for the constrained representation problem.